DE-A-102 11 664 and U.S. Pat. No. 6,822,068 disclose processes for preparing highly branched polyglycerols from glycidols using polyfunctional initiators having 1 to 10 000 hydroxyl, thiol and/or amino groups which must be deprotonated using suitable reagents before being used. Resultant by-products such as methanol and water must be removed by distillation prior to the actual polymerization. Thus prepared, the initiator system is then dissolved in a preferably amidated solvent, into which a solution of the monomer (glycidol), diluted with a further solvent, is metered at temperatures of 80 to 140° C.